U.S. patent number 8,969,831 [Application Number 13/768,725] was granted by the patent office on 2015-03-03 for excitation enhancement and extraction enhancement with photonic crystals.
This patent grant is currently assigned to Massachusetts Institute of Technology. The grantee listed for this patent is Massachusetts Institute of Technology. Invention is credited to Song-Liang Chua, John Joannopoulos, Jeongwon Lee, Ofer Shapira, Marin Soljacic, Bo Zhen.
United States Patent |
8,969,831 |
Shapira , et al. |
March 3, 2015 |
Excitation enhancement and extraction enhancement with photonic
crystals
Abstract
Disclosed herein is a system for stimulating emission from at
least one an emitter, such as a quantum dot or organic molecule, on
the surface of a photonic crystal comprising a patterned dielectric
substrate. Embodiments of this system include a laser or other
source that illuminates the emitter and the photonic crystal, which
is characterized by an energy band structure exhibiting a Fano
resonance, from a first angle so as to stimulate the emission from
the emitter at a second angle. The coupling between the photonic
crystal and the emitter may result in spectral and angular
enhancement of the emission through excitation and extraction
enhancement. These enhancement mechanisms also reduce the emitter's
lasing threshold. For instance, these enhancement mechanisms enable
lasing of a 100 nm thick layer of diluted organic molecules
solution with reduced threshold intensity. This reduction in lasing
threshold enables more efficient organic light emitting devices and
more sensitive molecular sensing.
Inventors: |
Shapira; Ofer (Cambridge,
MA), Soljacic; Marin (Belmont, MA), Zhen; Bo
(Cambridge, MA), Chua; Song-Liang (Cambridge, MA), Lee;
Jeongwon (Boston, MA), Joannopoulos; John (Belmont,
MA) |
Applicant: |
Name |
City |
State |
Country |
Type |
Massachusetts Institute of Technology |
Cambridge |
MA |
US |
|
|
Assignee: |
Massachusetts Institute of
Technology (Cambridge, MA)
|
Family
ID: |
51350266 |
Appl.
No.: |
13/768,725 |
Filed: |
February 15, 2013 |
Prior Publication Data
|
|
|
|
Document
Identifier |
Publication Date |
|
US 20140230884 A1 |
Aug 21, 2014 |
|
Current U.S.
Class: |
250/458.1;
250/459.1 |
Current CPC
Class: |
H01S
3/168 (20130101); H01L 31/055 (20130101); G01N
21/658 (20130101); G01N 21/648 (20130101); H01L
31/0547 (20141201); Y02E 10/52 (20130101); G02B
6/1225 (20130101); Y10S 977/774 (20130101); H01S
3/094034 (20130101); Y10S 977/759 (20130101); B82Y
20/00 (20130101); H01S 3/022 (20130101) |
Current International
Class: |
G01N
21/64 (20060101) |
Field of
Search: |
;250/458.1,459.1 |
Other References
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structure," Appl. Phys. Lett. 75, 316 (1999); doi:
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photonic crystals," Nature Materials, vol. 3, pp. 211-219 (Apr.
2004). cited by applicant .
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nonlinear phase sensitivity," J. Opt. Soc. Am. B., vol. 19, No. 9,
pp. 2052-2059 (Sep. 2002). cited by applicant .
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interaction," presentation at George Washington University on Feb.
16, 2012. cited by applicant .
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transmission at the Dirac point of a photonic band structure,"
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applicant .
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Review, vol. 70, pp. 235123-1-235123-7 (2004). cited by applicant
.
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surface-emitting lasers enabled by the accidental Dirac-point," in
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2012. cited by applicant .
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.
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|
Primary Examiner: Gaworecki; Mark R
Attorney, Agent or Firm: Cooley LLP
Government Interests
GOVERNMENT SUPPORT
This invention was made with government support under Contract Nos.
DE-SC0001299 and DE-FG02-09ER46577 awarded by the Department of
Energy, under Contract No. W911NF-07-D-0004 awarded by the Army
Research Office and under Grant No. DMR0819762 awarded by the
National Science Foundation. The government has certain rights in
the invention.
Claims
What is claimed is:
1. A system for stimulating an emission from at least one emitter,
the system comprising: a photonic crystal, characterized by an
energy band structure exhibiting a Fano resonance, comprising a
patterned dielectric substrate defining a surface to support the at
least one emitter; and a radiation source, in optical communication
with the photonic crystal and the at least one emitter, to
irradiate the at least one emitter at a first angle with respect to
the surface of the photonic crystal so as to stimulate the emission
from the at least one emitter at a second angle with respect to the
surface of the photonic crystal.
2. The system of claim 1, wherein the at least one emitter
comprises at least one of an organic molecule, a quantum dot, an
organic quantum dot, a quantum well, and an exciton-hole pair.
3. The system of claim 1, wherein the photonic crystal has a
quality factor of about 10 to about 10.sup.10.
4. The system of claim 3, wherein the quality factor extends over
about 10.sup.2 unit cells of the photonic crystal to about
10.sup.10 unit cells of the photonic crystal.
5. The system of claim 1, wherein the patterned dielectric
substrate defines a plurality of cylindrical holes arrayed on a
square lattice.
6. The system of claim 1, wherein the at least one emitter emits
substantially all of the emission at the second angle with respect
to the surface of the photonic crystal.
7. The system of claim 1, further comprising: a detector, in
optical communication with the at least one emitter, to sense the
emission.
8. The system of claim 7, wherein the emission comprises a
fluorescence signal and the detector is configured to detect the
fluorescence signal.
9. The system of claim 7, wherein: the at least one emitter
comprises at least one organic molecule characterized by a Raman
resonance frequency; the Fano resonance is at a Fano resonance
frequency substantially equal to the Raman resonance frequency; the
radiation source is configured to excite the at least one organic
molecule with coherent radiation at the Fano resonance frequency so
as to cause enhancement of the emission via resonant absorption of
the coherent radiation by the photonic crystal.
10. The system of claim 9, wherein the detector comprises a
spectrometer to measure a spectrum of the emission.
11. The system of claim 9, further comprising: a channel to guide a
solution containing the at least one organic molecule over a
surface of the photonic crystal; and a reservoir, in fluid
communication with the channel, to store the solution containing
the at least one organic molecule.
12. A method of stimulating an emission from at least one emitter,
the method comprising: (A) supporting the at least one emitter of a
surface of a photonic crystal, characterized by an energy band
structure exhibiting a Fano resonance, comprising a patterned
dielectric substrate free of defects; and (B) irradiating the at
least one emitter at a first angle with respect to the surface of
the photonic crystal so as to stimulate the emission from the at
least one emitter at a second angle with respect to the surface of
the photonic crystal.
13. The method of claim 12, wherein (A) further comprises: flowing
a solution containing the at least one emitter over the surface of
the photonic crystal.
14. The method of claim 12, wherein (B) further comprises:
selecting the first angle so as to cause the at least one emitter
to emit substantially all of the emission at the second angle with
respect to the surface of the photonic crystal.
15. The method of claim 12, wherein (B) further comprises:
selecting a frequency at which to irradiate the at least one
emitter based on (i) a Fano resonance frequency of the Fano
resonance and (ii) a resonance frequency of the at least one
emitter so as to resonantly enhance the emission via resonant
absorption by the photonic crystal.
16. The method of claim 12, further comprising: (C) detecting the
emission.
17. The method of claim 16, wherein the emission comprises at least
one of fluorescence, phosphorescence, and a Raman signal emitted by
the at least one emitter and (C) comprises detecting the at least
one of the fluorescence, the phosphorescence, and the Raman
signal.
18. The method of claim 16, further comprising: (D) determining at
least one characteristic of the at least one emitter based in part
on the emission detected in (C).
19. The method of claim 18, wherein (D) further comprises:
determining a spectrum of the emission; and identifying the at
least one emitter based at least in part on the spectrum.
Description
BACKGROUND
Organic molecules are pervasive in daily life: from natural
proteins, to human synthesized fluorescing labels, to organic
semiconductors. The interaction of light with such molecules is at
the heart of important technological advances in biomolecular
detection, fluorescent microscopy, and organic light emitting
devices as well as more fundamental studies of cavity quantum
electrodynamics and various types of enhanced spectroscopy and
sensing. This interaction can be altered or enhanced by placing the
organic molecules in a nanostructured cavity where both the
lifetime of the resonances and the optical density of states (DOS)
can be tailored.
However, there are inherent challenges in incorporating organic
molecules in such cavities: first, their dissimilar compositional
structure makes it difficult to incorporate them within the high
dielectric regions of the cavity where long-lifetime resonances
concentrate their electromagnetic energy. Second, micro- and
nanostructured cavities typically only have a small portion of
their model volumes extending outside their high-dielectric
regions, making it challenging to bring external entities precisely
to within that volume. Third, it is difficult to pattern organic
materials at the nano-scale; indeed, organic patterning processes
tend to be incompatible with inorganic processes. These challenges
limit experimental realizations of systems of excitons of organic
molecules and optical resonances compared to systems of inorganic
quantum nano structures.
SUMMARY
Embodiments of the present invention include a system and
corresponding method for stimulating emission from at least one
emitter, such as an organic molecule, a quantum dot, an organic
quantum dot, a quantum well, or an exciton-hole pair. This system
may include a photonic crystal and a radiation source. The photonic
crystal, which is characterized by an energy band structure
exhibiting a Fano resonance, includes a patterned dielectric
substrate free of defects and defines a surface to support the
source. The radiation source irradiates the source at a first angle
with respect to the surface of the photonic crystal so as to cause
the source to emit radiation at a second angle with respect to the
surface of the photonic crystal. In some examples, the first angle
is selected so as to cause the source to emit substantially all of
the radiation at the second angle with respect to the photonic
crystal's surface.
In at least one embodiment, the photonic crystal has a quality
factor of about 10 to about 10.sup.10. This quality factor may
extend over about 10.sup.2 unit cells of the photonic crystal to
about 10.sup.10 unit cells of the photonic crystal. The photonic
crystal's patterned dielectric substrate defines a plurality of
cylindrical holes arrayed on a square lattice.
Embodiments may also include a detector, in optical communication
with the one source, to sense the power emitted at the second angle
by the source. This detector may be configured to detect
fluorescence, phosphorescence, and/or a Raman signal emitted by the
source.
In certain embodiments, the detector may be configured to detect an
enhanced Raman signal emitted by the source. In these embodiments,
the source comprises at least one organic molecule characterized by
a Raman resonance frequency, which is substantially equal to the
resonance frequency of the Fano resonance. The radiation source
excites the organic molecule with coherent radiation at the Fano
resonance frequency so as to cause the radiation emitted by the
organic molecule to be enhanced via resonant absorption of the
coherent radiation by the photonic crystal. The detector, which may
include a spectrometer, senses at least one characteristic of this
enhanced radiation. For instance, the detector may determine the
enhanced radiation's spectrum, which can be used to identify the
source.
In some cases, the system also includes a channel and a reservoir
in fluid communication with the surface of the photonic crystal.
The reservoir stores organic molecules or other sources in
solution. This solution flows through the channel the photonic
crystal's surface, where it is irradiated and its emission
(fluorescence, phosphorescence, or Raman signal) is detected.
Embodiments of the present invention also include a solar
concentrator that comprises a photonic crystal and solar cell. The
photonic crystal, which is characterized by an energy band
structure exhibiting a Fano resonance, absorbs radiation incident
over a first solid angle on a first surface of the photonic crystal
and emit radiation over a second solid angle that is smaller than
the first solid angle via a second surface of the photonic crystal.
The solar cell, which is in optical communication with the photonic
crystal's second surface, receives at least a portion of the
radiation emitted by the photonic crystal via the photonic
crystal's second surface. In some embodiments, the photonic crystal
comprises at least one layer of dielectric material having a
plurality of cylindrical holes arrayed on a square lattice. The
solar concentrator may also include a frequency converter, in
optical communication with the photonic crystal, that shifts a
frequency of the incident radiation to an absorption band of the
photonic crystal.
All combinations of the foregoing concepts and additional concepts
discussed in greater detail below (provided such concepts are not
mutually inconsistent) are part of the inventive subject matter
disclosed herein. In particular, all combinations of claimed
subject matter appearing at the end of this disclosure are part of
the inventive subject matter disclosed herein. Terminology
explicitly employed herein that also may appear in any disclosure
incorporated by reference should be accorded a meaning most
consistent with the particular concepts disclosed herein.
BRIEF DESCRIPTION OF THE DRAWINGS
The skilled artisan will understand that the drawings primarily are
for illustrative purposes and are not intended to limit the scope
of the inventive subject matter described herein. The drawings are
not necessarily to scale; in some instances, various aspects of the
inventive subject matter disclosed herein may be shown exaggerated
or enlarged in the drawings to facilitate an understanding of
different features. In the drawings, like reference characters
generally refer to like features (e.g., functionally similar and/or
structurally similar elements).
FIG. 1A is a schematic diagram of a photonic crystal with an energy
band structure that exhibits a Fano resonance.
FIG. 1B is a diagram of the energy band structure of the photonic
crystal shown in FIG. 1A.
FIG. 1C is a energy level diagram of an organic molecule on the
surface of the photonic crystal shown in FIG. 1A.
FIG. 1D illustrates a photonic crystal with linear defects (left)
and its photonic band structure (right).
FIG. 1E illustrates another photonic crystal with linear defects
(left), its guided mode (center), and its photonic band structure
(right).
FIG. 1F illustrates a multiply periodic photonic crystal (left) and
its photonic band structure (right).
FIG. 1G illustrates a uniformly periodic photonic crystal (left)
and its photonic band structure (right).
FIG. 1H is a plot of the Purcell enhancement for photonic crystals
with ultra-flat bands (upper curve) and coupled-cavity modes (lower
curve).
FIGS. 1I, 1J, and 1K shows plots of quantum yield for flat-band,
coupled-defect, plasmonic, and conventional photonic crystals for
quality factors of 10, 100, and 1000, respectively.
FIG. 2A illustrates a system for enhancing radiation emitted by a
source flowing over the surface of the photonic crystal shown in
FIG. 1A.
FIG. 2B illustrates a solar concentrator using the photonic crystal
shown in FIG. 1A.
FIG. 3 shows an apparatus suitable for fabricating a photonic
crystal with an energy band structure that exhibits a Fano
resonance.
FIGS. 4A-4D illustrate a process of fabricating a photonic crystal
with an energy band structure that exhibits a Fano resonance.
FIG. 4E shows scanning electron microscopes images of a photonic
crystal fabricated according to the process shown in FIGS.
4A-4D.
FIG. 5 shows band diagrams of the photonic crystal in FIG. 4
obtained from reflectivity measurements with (a, b) E.sub.y and (d,
e) E.sub.x polarized beams and finite difference time domain (FDTD)
simulations for (c) E.sub.y and (f) E.sub.x polarized beams.
FIG. 6 is a plot of simulated radiative quality factor versus angle
for high-Q singly-degenerate modes (solid lines) and
doubly-degenerate modes (dotted lines).
FIG. 7 is a plot of fitted quality factors versus angle for the
measured data plotted in FIG. 5 (the left-hand insets show the
reflectivity spectra of leaky mode 5 measured at 0.1.degree.,
0.4.degree., and 0.8.degree.; the right-hand inset depicts an
example of the curve fitting process).
FIG. 8 is a plot of intensity versus wavelength for radiation
emitted by a uniform collection of randomly polarized dipoles
generated by excited organic molecules on the surface of an
inventive photonic crystal (uniform slab) illuminated with
on-resonant and off-resonant excitation radiation.
FIG. 9 illustrates simulated and experimental results of total
emission enhancement from a Rhodamine 6G (R6G) solution on the
surface of an inventive photonic crystal, including: (a) the
photonic crystal's band structure along the .GAMMA. to M and
.GAMMA. to X directions; (b) angle-resolved fluorescence
measurements of the R6G solution; (c) predictions of the total
enhancement factors vents angle for modes 1 and 4; (d) predicted
average total enhancement factor versus angle from modes 1 and 4;
and (e) total enhancement factor extracted from experimental
results in (b).
FIG. 10 is a plot of output energy measured with a spectrometer
(open circles), output energy measured with a power meter (filled
circles), and predicted output energy (line) versus pump energy for
a laser made of a solution of organic molecules on the surface of a
photonic crystal whose energy band structure exhibits.
DETAILED DESCRIPTION
Following below are more detailed descriptions of various concepts
related to, and embodiments of, inventive systems, methods, and
apparatus for enhancing emissions from sources such as organic
molecules, quantum dots, quantum wells, etc. The various concepts
introduced above and discussed in greater detail below may be
implemented in any of numerous ways, as the disclosed concepts are
not limited to any particular manner of implementation. Examples of
specific implementations and applications are provided primarily
for illustrative purposes.
One example includes a system with a dielectric surface that
enables simple incorporation of organic molecules onto a
nanostructured resonant cavity. This system demonstrates strongly
enhanced interaction of light with organic molecules that are
brought to within one hundred nanometers from its macroscopic
interface. The dielectric surface, which is patterned with a
sub-wavelength, periodic structure, supports a special type of Fano
resonance with wave functions extending above it. The delocalized
nature of these resonances, their long lifetimes, and the
structure's altered spectral density of states (SDOS) causes
changes in the organic molecule's spectral and angular radiation
pattern compared to the molecule's free-space emission pattern.
Placing molecules close to the surface yields sharp spectral
features in the molecules' fluorescence spectra, with an
enhancement of the differential radiating power (related to
brightness) by a factor of up to 6.3.times.10.sup.3. Without being
bound by any particular theory, it appears that this enhancement
can be attributed to two mechanisms: (1) enhancement of the local
excitation field through coupling to a resonance mode in the
photonic crystal and (2) enhancement of extraction rate of emitted
photons in the far field. A theoretical model, derived below from
coupled mode theory (CMT) and Green functions expansion in the
basis of Bloch modes, can be used to predict the contribution of
each mechanism to the total enhancement. Furthermore, the two
enhancement mechanisms also contribute to reduce the lasing
threshold by at least an order of magnitude when compared to
previous experiments with similar molecules. Photonic crystals
exhibiting this special type of Fano resonances can also be used to
support lasing by organic dye molecules as discussed in greater
detail below.
As understood by those of skill in the art, a Fano resonance is a
resonance that arises from interference of a narrow discrete
resonance with a broad spectral line or continuum. More
specifically, the intensity transmitted or reflected by a Fano
resonance exhibits an asymmetric shape with the following
functional form:
.varies..times..times..omega..omega..omega..omega. ##EQU00001##
where .omega..sub.0 and .nu. are standard parameters that denote
the position and width of the resonance, respectively, and
|F|.ltoreq.1 describes the degree of asymmetry. The Fano resonances
in the photonic crystals disclosed here each have a maximum peak
and a minimum trough (e.g., as shown in FIGS. 5(b) and (e)).
In photonic crystal slabs, the physical origin of Fano resonances
lies in the coupling between the guided modes supported by the slab
and external plane waves, which occurs because of the periodic
modulation of the dielectric constant. The modes supported by
photonic crystals generally fall into two categories: (1) pure
modes with infinite lifetimes that lie outside the light cone and
(2) resonant modes with finite lifetimes that lie within the light
cone and consequently can couple to radiation modes. These
lifetimes may also be expressed as quality factors (also known as
"Q factors" or simply "Q"), which are measures of how slowly a
resonance dissipates energy or, equivalently, how long the
resonance stores energy.
There is a special subset of Fano resonances whose quality factors
may approach infinity. In theory, in a perfect infinite periodic
photonic crystal slab, due to symmetry considerations, Fano
resonances at a wave vector of k=0 may completely decouple from the
external world with infinite radiative quality factor (Q.sub.rad)
despite lying within the light cone. For k near zero, these guided
resonances have ultra-long (but finite) lifetimes, providing an
efficient means to couple light in and out of the slab.
Because the guided resonances in these photonic crystals have such
long lifetimes, they can be used to resonantly enhance absorption
by sources on the surfaces of these photonic crystals. To see how,
consider a source, such as an organic molecule or quantum dot, on
the surface of such a photonic crystal. If the source has a
resonance whose frequency substantial coincides with the photonic
crystal's Fano resonance frequency, then the source will excite the
photonic crystal's resonance mode, resulting in large local field
enhancement near the photonic crystal's surface. This enhanced
local field results in increased absorption by the source. In other
words, a photonic crystal with a Fano resonance at or near k=0
provides excitation enhancement of incident radiation for the
source.
A photonic crystal with a Fano resonance also provides extraction
enhancement of the radiation emitted by the source. Suppose that
the source radiates in response to absorption of incident
radiation--for example, it may fluoresce, phosphoresce, or emit a
Raman signal. In free space, the source emits this radiation
isotropically. When electromagnetically coupled to the photonic
crystal and excited from a first direction, the source radiates
preferentially in a second direction instead of radiating
isotropically. As a result, the radiant intensity (power per solid
angle) goes up in the preferentially illuminated direction. (The
radiant intensity goes down in other directions to conserve
energy.) Thus, the photonic crystal's Fano resonance causes an
angular redistribution of radiation emitted by a source on the
crystal's surface. Fano resonances at or near k=0 tend to have high
quality factors, leading to even larger enhancement values.
Until now, experimental verification of high-Q Fano resonances at
or near k=0 over a macroscopically large area had yet to be
demonstrated, possibly because of photonic crystal fabrication and
material challenges. One challenge in observing these resonances is
that in practical structures, in addition to limits imposed by
material absorption, fabrication imperfections may break the
crystal symmetry, which results in coupling of these Fano
resonances to radiating modes. In addition, extending the mode over
a macroscopic area in order to support a higher radiative quality
factor poses a significant fabrication challenge.
Photonic Crystals with Fano Resonances for Emission Enhancement
FIG. 1A is a schematic diagram of a large-area, square-lattice
photonic crystal 100 whose energy band structure (shown in FIG. 1B)
includes a Fano resonance at or near a wave vector of k=0 (i.e., at
or near the .GAMMA. point). Unlike other photonic crystals with
Fano resonances, this photonic crystal 100 is free of defects.
Instead, it includes a dielectric substrate 110 with a large area
that is uniformly patterned with holes 112 arrayed on a square
lattice. FIG. 1A also shows the Fano resonance's egg-crate-like
energy density surface 101, which has troughs over the holes 112
and peaks over the dielectric material 114 surrounding the holes
112.
In one example, the substrate 110 includes a 250 nm thick slab of
Si.sub.3N.sub.4 with periodic cylindrical holes 112 on top of 6
.mu.m thick SiO.sub.2 layer. The holes 112 are spaced at an average
period of 320 nm, with an average hole diameter of 160 nm and an
average hole depth of 55 nm. These uniformly periodic hole patterns
may extend over several square centimeters (e.g., 1, 2, 3, 4, or 5
cm.sup.2). Those of ordinary skill in the art will readily
appreciate that other hole spacings, diameters, and depths are
possible, as are slabs of other materials or thicknesses.
The dielectric substrate 110 defines a surface 112 to hold one or
more resonant sources, such as organic molecules, quantum dots
(including organic and inorganic quantum dots), quantum wells, and
exciton-hole pairs. In this case, the dielectric substrate 110
supports several emitters--here, organic dye molecules 10--each of
which is characterized by the energy level diagram shown in FIG.
1C. For instance, the organic dye molecules 10 may be Rhodamine 6G
(R6G) dissolved in methanol at 1 mM concentration and placed on top
of the photonic crystal 100. This energy level diagram shows that
irradiating one of these dye molecules 10 with pump radiation from
a radiation source, which may be a coherent light source (e.g., a
laser 120) or an incoherent light source (e.g., a light-emitting
diode, a white-light source, a supercontinuum source, etc.), causes
the dye molecule 10 to undergo a transition from a lower singlet
state S.sub.0 to a higher singlet state S.sub.1. The excited dye
molecule 10 relaxes back to the lower singlet state S.sub.0 through
stimulated emission, spontaneous emission, non-radiative
transitions, and vibrational transitions. It may also reabsorb the
emitted radiation.
A detector, such as a spectrometer 130, senses the radiation (e.g.,
fluorescence) emitted by the organic molecule(s) illuminated by the
laser 120. The spectrometer 130 may use the detected signal to
determine the fluorescence spectrum of the organic molecule(s),
which in turn can be used to identify the organic molecules 10. For
instance, the photonic crystal 100, laser 120, and spectrometer 130
may be used for fluorescence spectroscopy or Raman spectroscopy:
the laser 120 illuminates the organic molecule 10 at the
appropriate frequency, causing the molecule 10 to emit fluorescent
light or a spontaneous or stimulated Raman scattering signal. The
spectrometer 130 determines the spectrum of this emitted light; as
understood by those of skill in the art, this spectrum may be used
to identify the molecule.
As mentioned above, the photonic crystal 100 enhances the signal
emitted by the organic molecule 10. This enhancement results in an
increase in the intensity of the signal measured by the detector
130 (assuming that the laser 120 and the detector 130 are properly
aligned). Without being bound by any particular theory, it appears
that two mechanisms provide this enhancement: (1) the photonic
crystal's resonant absorption and subsequent dissipation of the
incident laser light provides excitation enhancement and (2) the
photonic crystal's modified spectral density of states provides
extraction enhancement by restricting the molecule's fluorescence
emission to a relatively small solid angle.
Excitation enhancement occurs in structures that support resonances
for the excitation wavelength via enhancement of the local electric
field in the site of the molecules. In many nanostructured
resonances, the active volume of the organic material that
interacts with the resonance is small (compared to the wavelength),
so only a small fraction of the excitation beam is absorbed.
However, the local excitation field can be orders of magnitude
higher than in free space when the pump is coupled to resonances
with long lifetimes (the pump resonant modes). This coupling leads
to enhanced absorption. The power absorbed by bulk molecules is
given by P.sup.B.sub.abs=(N.sub.0.sigma..sub.absd)P.sub.in, where
.sigma..sub.abs is the absorption cross-section of molecules at the
excitation wavelength, N.sub.0 is the number density of molecules,
d is the thickness of the layer that the molecules occupy and
P.sub.in is the pump power. Through coupled mode theory, the
absorption enhancement in a layer of thickness d.sup.P.sub.eff
coupled to a resonant pump mode compared to bulk absorption is
given by:
.LAMBDA..ident..times..lamda..pi..times..times..times..alpha..function.
##EQU00002## where .lamda..sup.P is the pump wavelength, n is the
refractive index of the liquid in which the organic molecules are
dissolved, Q.sup.P.sub.r and Q.sup.P are the radiative and total
quality factors, respectively, of the pump mode, d.sup.P.sub.eff is
the length of the evanescent tail of the pump mode into the
molecule layer, and .alpha..sup.P is the energy confinement of the
pump mode in the molecule layer. The quantities in Equation (1) can
be found either by finite-difference time-domain (FDTD) simulation
or reflection measurements. The maximum extraction enhancement
occurs when the Q-matching condition between the radiative and
non-radiative quality factors is satisfied.
Extraction enhancement is due to the strong modification of the
spectral density of states (SDOS) in the presence of a Fano
resonance. Coupling the molecules to a macroscopic nanostructure
resonance dramatically alters the molecules' angular emission
compared to free-space emission. When coupled to a resonance, the
rate at which a uniform and isotropic collection of molecules
generates photons with crystal momentum k at a resonant frequency
.omega..sub.k can be written as:
.GAMMA..function..omega..times..pi..times..times..omega..times..mu..times-
. .epsilon..times..alpha..function..omega..times. .function..omega.
##EQU00003##
This result can be achieved by expanding the Green function with a
basis of normalized Bloch modes, E.sub.k.omega..sub.--.sub.k(r),
with finite lifetimes characterized by the total quality factor
Q.sup.F.sub.tot(k,
.omega..sub.k)=.omega..sub.k/(2.DELTA..omega..sub.k). Here,
.alpha..sup.F.sub.k,.omega..sub.--.sub.k is the energy confinement
of the fluorescence resonance mode in the molecule layer, S(k,
.omega..sub.k)=A/(4.pi..sup.3.DELTA..omega..sub.k) is the SDOS at
the resonance .omega..sub.k crystal momentum k, A is the area of
the macroscopic Fano resonance, and |.mu.| is the electric dipole
momentum of the molecules. The extraction enhancement into the far
field when on photonic crystal compared to in free-space under the
assumption that the radiation direction is close to normal
direction can be written as:
.LAMBDA..function..omega..GAMMA..times..GAMMA..lamda..times..alpha..times-
..times..pi..times..times..times. ##EQU00004## where
d.sup.F.sub.eff(k, .omega..sub.k) is the effective length of the
evanescent tail of the fluorescence mode in the molecule layer and
Q.sup.F.sub.r(k, .omega..sub.k) is the radiative quality factor of
the fluorescing channel. Like the quantities in Equation (1), the
quantities in Equation (3) can be obtained from FDTD calculations
and reflection measurements.
Equation (3) shows that increasing enforcing the Q-matching
condition of Q.sub.nr(k, .omega..sub.k)=Q.sup.tot.sub.r(k,
.omega..sub.k) increases the extraction enhancement, just it
increases the excitation enhancement. Increasing the energy
confinement of the fluorescence resonance mode in the molecule
layer also increases the extraction enhancement.
There are three major differences between this formalism and local
density of states (LDOS) enhancement calculations in micro-cavity
systems: (1) this formalism deals with the emission from a uniform
and isotropic ensemble of molecules placed on a periodic
macroscopic photonic crystal into a fixed crystal momentum k at
.omega..sub.k, which is proportional to the system's SDOS instead
of to the LDOS (which is proportional to the emission of one dipole
into all directions); (2) this formalism treats an infinitely large
system by expanding Green functions with a basis of Bloch modes
under periodic boundary condition instead of localized eigenmodes
as often used in LDOS calculations; and (3) this formalism accounts
for the photons that are radiated coherently to the far field and
reach. As a result, the maximizing condition changes from
maximizing Q.sub.tot in general to enforcing the Q-matching
condition.
Given knowledge of the local excitation and extraction enhancement,
the total enhancement factor can be approximated as the product of
the excitation and extraction enhancement factors:
.LAMBDA..function..omega..times..eta..times..times..eta..times..times..la-
mda..times..lamda..pi..times..times..times..times..times..times..times..ti-
mes..intg..times..function..times..function..times..times.d
.times..times..apprxeq..times..LAMBDA..times..LAMBDA..function..omega.
##EQU00005## This approximation is valid under two conditions: (1)
the quantum yield of the molecules remains constant, and (2) the
normalized pump and fluorescence mode profiles are roughly
uniformly distributed in a similar region in space, meaning the
overlap integral in Equation (4) can be simplified as the product
of the fraction of pump mode energy in the molecule layer and the
energy confinement of the fluorescence resonance mode in the
molecule layer. The latter approximation is commonly ignored, but
can lead to further enhancement.
These two enhancement mechanisms may also reduce the lasing
threshold of a source (e.g., one or more organic molecules) on the
surface of a photonic crystal for at least two reasons. First, the
excitation field is dramatically enhanced near the surface of the
photonic crystal. This enables substantial absorption of the pump
within a thin layer of diluted molecules near the photonic crystal
surface. Second, placing the molecules on the photonic crystal's
surface enhances the molecules' emission rate into the lasing mode
compared to their free-space emission in a similar modal volume.
This enhancement can be introduced phenomenologically into the
lasing rate equation through the spontaneous emission factor,
.beta., which is classically defined as the ratio between the
emission rate into the lasing mode and the total emission rate. The
lasing threshold is typically inversely proportional to .beta. and
hence can be reduced in cases where the emission rate into the
lasing mode is enhanced while the total rate remains almost
constant.
Photonic Crystals with Fano Resonances for Quantum Yield
Enhancements
Apart from on-resonance coupling effects, the radiative decay rates
of a molecule placed on the surface of an inventive photonic
crystal may be altered significantly while coupled to a Fano
resonance supported by the inventive photonic crystal. Therefore,
the molecule's far field emission signal is stronger and its
quantum yield is greater on the photonic crystal's surface than on
the surface of a bulk dielectric material.
The increase in the averaged enhancement of the radiative decay
rate and quantum yield depends on the photonic crystal's band
structure, the signals' frequency distribution, the original
quantum yield enhancement, etc. For a uniform collection of
randomly polarized dipoles placed on top of a photonic crystal
surface, the average enhancement of radiative decay rate (also
known as the local Purcell enhancement) can be estimated as:
.function..ident..GAMMA..GAMMA..times..pi..times..omega..DELTA..times..ti-
mes..omega..times..function. ##EQU00006## where .omega. is the
center frequency of the signal, which is also assumed to be the
center of the resonance frequency;
.DELTA..omega.=max{.DELTA..omega..sub.R; .DELTA..omega..sub.S}
where .DELTA..omega..sub.R is the linewidth of resonance and
.DELTA..omega..sub.S is the linewidth of the source signal; and
.function. ##EQU00007## is the local field enhancement due to the
resonance. With this in mind, the spatially averaged Purcell
enhancement can be written as:
.ident..GAMMA..GAMMA..apprxeq..times..times..pi..times..function..DELTA..-
times..times..omega..times..times..pi..times..omega..DELTA..times..times..-
omega..times..alpha..times..times..lamda..times. ##EQU00008## where
.lamda. is the center wavelength of the source and the resonance;
d.sub.eff is the effective length of the evanescent tail of the
resonance; a is the photonic crystal's periodicity; and
.function..DELTA..times..times..omega..times..pi. ##EQU00009## is
the portion of one full Brillouin zone with photonic crystal
resonance frequencies within the source's signal range.
To increase the spatially averaged Purcell enhancement, one can:
(1) increase the portion of the Brillouin zone within the source's
signal range; (2) match the center frequency of resonance to that
of the source; and (3) match the quality factor of the
resonances
.omega..DELTA..times..times..omega. ##EQU00010## to the quality
factor of the source
.omega..DELTA..times..times..omega. ##EQU00011## Accordingly, the
quantum yield enhancement can be written as:
.LAMBDA..eta..ident..eta..eta..eta..times..eta. ##EQU00012## This
expression can be used to estimate the quantum yield enhancement
for different photonic structures as explained below.
FIG. 1D shows a photonic crystal 150 (left) comprising a square
lattice of high-.di-elect cons. dielectric rods 152 (.di-elect
cons..sub.H=12.25) embedded in a low-.di-elect cons. dielectric
material (.di-elect cons..sub.L=2.25). The lattice spacing is
denoted a, and the radius of each rod is r=0.25a. This photonic
crystal is known as a coupled-cavity waveguide, or coupled
resonator optical waveguides, because it includes many cavities 154
formed by reducing the radius of every fourth rod along one row of
the array to r/3 to give a supercell period 4a. When isolated, each
of these cavities supports a resonant mode with a resonant
frequency well inside the bandgap. Bringing such cavities close to
each other to form a linear defect, as shown in FIG. 1D, enables
photons to propagate down the defect by tunneling from one cavity
to another. Consequently, the group velocity is small; the less
closely coupled the cavities are, the slower the group velocity.
Group velocities of c/1000 or even smaller are easy to attain in
such systems.
The plot at right in FIG. 1D shows the induced change in the
photonic band frequency for the CCW photonic crystal. In
particular, it shows the effect of induced refractive index changes
for two dispersion curves: the slow-light band with
.nu..sub.G=0.022c used in the photonic crystal at left (sine-like
solid curve) and the dispersion curve of a uniform material with
n=3.5 (nearly vertical solid curve). Applying the same frequency
shift (dv 5 0.001) to both dispersion curves yields the respective
dashed curves, which exhibit different changes in wave vector for
identical frequency shifts.
FIG. 1E shows another CCW photonic crystal 160 (left) formed by
patterning holes 162 in a dielectric slab 161. Holes 164 arrayed
along one column are enlarged to form a coupled-cavity waveguide.
The central image in FIG. 1E illustrates the guided mode, and the
plot at right illustrates the CCW photonic crystal's photonic band
structure.
The averaged Purcell enhancement and quantum yield enhancement for
the photonic crystals in FIGS. 1D and 1E can be written,
respectively, as:
.ident..GAMMA..GAMMA..apprxeq..times..times..pi..times..function..DELTA..-
times..times..omega..times..times..pi..times..omega..DELTA..times..times..-
omega..times..alpha..times..times..lamda.<.times..times..pi..times..tim-
es..omega..DELTA..times..times..omega..times..alpha..times..times..lamda.
##EQU00013## .LAMBDA..eta..ident..eta..eta..eta..times..eta.
##EQU00013.2## where L.sub.k(.DELTA..omega.) is the length in 1D
k-space with frequency within the signal frequency range and N is
the number of basic unit cells in the super-cell.
FIG. 1F shows a coupled resonator photonic crystal 170 with holes
172 on a first square lattice of period a in a dielectric slab 171.
This photonic crystal 170 also includes filled holes 174 arranged
on second square lattice of period 3a that is aligned with the
first square lattice. The plots at right in FIG. 1F illustrate
experimental measurements of this photonic crystal's band
structure. The averaged Purcell enhancement and quantum yield
enhancement for the photonic crystal 170 in FIG. 1F can be written,
respectively, as:
.ident..GAMMA..GAMMA..apprxeq..times..times..pi..times..function..DELTA..-
times..times..omega..times..times..pi..times..omega..DELTA..times..times..-
omega..times..alpha..times..times..lamda..times.<.times..times..pi..tim-
es..times..omega..DELTA..times..times..omega..times..alpha..times..times..-
lamda..times. ##EQU00014##
.LAMBDA..eta..ident..eta..eta..eta..times..eta. ##EQU00014.2##
where N is the number of basic unit cells in the super-cell (the
photonic crystal 170 shown in FIG. 1F includes nine basic unit
cells per super-cell).
FIG. 1G shows a photonic crystal 180 formed by square lattice of
rods 182 with an index of refraction n=3.4 and radius r=0.134a. The
plot at the center of FIG. 1G is the p projected transverse
magnetic band structure of the photonic crystal 180. The black
lines represent .omega.(k.sub.x) curves at fixed k.sub.y values
ranging from k.sub.y=0 to k.sub.y=0.5(2.pi./a) in steps of
0.05(2.pi./a). The thick arrow indicates an ultra-flat
cross-section of the second band. The contour plot at right in FIG.
1G illustrates of the second band .omega.(k.sub.x, k.sub.y). The
saddle point and its principal axes are indicated by the black dot
at the center of the plot. The extended ultra-flat band is along
the dashed line.
For a flat band (e.g., a band whose variation with frequency is
less than or equal to about 20% to about 30% over the Brillouin
zone), the averaged Purcell enhancement and quantum yield
enhancement can be written, respectively, as:
.ident..GAMMA..GAMMA..apprxeq..times..times..pi..times..function..DELTA..-
times..times..omega..times..times..pi..times..omega..DELTA..times..times..-
omega..times..alpha..times..times..lamda..times. ##EQU00015##
.LAMBDA..eta..ident..eta..eta..apprxeq..eta..times..eta.
##EQU00015.2## For the flat second band in FIG. 1G, this yields a
Purcell enhancement of about 1.3, which is also close to quantum
yield enhancement.
For a 1D grating photonic crystal with a saddle point in its band
structure (e.g., the saddle point dispersion shown in FIG. 1G), the
estimated averaged Purcell enhancement and quantum yield
enhancement are:
.ident..GAMMA..GAMMA..apprxeq..times..times..pi..times..function..omega..-
DELTA..times..times..omega..times..alpha..times..times..lamda..times.
##EQU00016##
.LAMBDA..eta..ident..eta..eta..apprxeq..eta..times..eta.
##EQU00016.2## For a general photonic crystal with a flat-band
structure, the enhancements can be written as:
.ident..GAMMA..GAMMA..apprxeq..times..times..pi..times..function..DELTA..-
times..times..omega..times..times..pi..times..omega..DELTA..times..times..-
omega..times..alpha..times..times..lamda..times..ltoreq..times..times..pi.-
.times..omega..DELTA..times..times..omega..times..alpha..times..times..lam-
da..times. ##EQU00017##
.LAMBDA..eta..ident..eta..eta..apprxeq..eta..times..eta.
##EQU00017.2##
FIGS. 1H-1K are plots of the final averaged Purcell enhancement and
final quantum yield derived by defining different starting
conditions of
.omega..DELTA..times..times..omega..omega..DELTA..times..times..omega..om-
ega..DELTA..times..times..omega. ##EQU00018## and .eta..sub.0.
Here,
.omega..DELTA..times..times..omega. ##EQU00019## represents the
flatness of the band, with .DELTA..omega..sub.F being the largest
frequency deviation from the center frequency within the Brillion
zone.
The extraction enhancement and excitation enhancement are
independent of each other. The expressions above are for extraction
enhancement. Combining these extraction expressions with excitation
enhancement expressions yields the total enhancement:
.LAMBDA..intg..times..LAMBDA..function..fwdarw..times..LAMBDA..eta..funct-
ion..fwdarw..times..times.d.fwdarw..times. ##EQU00020##
.LAMBDA..function..fwdarw..times..times..lamda..pi..times..times..times..-
times..function..fwdarw. ##EQU00020.2## when the photonic crystal
is pumped on-resonance. Here, Q.sub.R is the quality factor of the
resonances; Q.sub.r is the radiative quality of the resonance; and
ratio Q.sub.R/Q.sub.r represents the averaged chance of generated
photons can reach far field while coupled to the photonic crystal
resonances.
Photonic Crystals with Fano Resonances for Raman Scattering
Enhancements
Photonic crystals with flat dispersions can also be use to enhance
Raman scattering signals. One difference between fluorescence and
Raman scattering enhancement is that the fluorescence is a
multi-step process, whereas Raman scattering is an instantaneous
process. Therefore, increasing the radiative decay rate may not
affect absorption cross-section for fluorescence; however, it may
affect the Raman scattering cross section. For similar multi-step
processes, the total enhancement may be predicted as above, whereas
the total enhancement for instantaneous processes can be written as
follows:
.function..ident..GAMMA..GAMMA..times..times..pi..times..omega..DELTA..ti-
mes..times..omega..times..function. ##EQU00021##
.LAMBDA..function..fwdarw..times..times..lamda..pi..times..times..times..-
times..function..fwdarw. ##EQU00021.2##
.LAMBDA..intg..times..LAMBDA..function..fwdarw..times..function..fwdarw..-
times..times.d.fwdarw..apprxeq..LAMBDA..times. ##EQU00021.3##
.ident..GAMMA..GAMMA..apprxeq..times..times..pi..times..function..DELTA..-
times..times..omega..times..times..pi..times..omega..DELTA..times..times..-
omega..times..alpha..times..times..lamda..times. ##EQU00021.4## The
fluorescence examples in the preceding section can also be used to
estimate Raman scattering enhancement by replacing the appropriate
parameters in the above equations.
Photonic Crystal-Based Raman Spectroscopy System
FIG. 2A illustrates a spectroscopy system 200 based on the photonic
crystal 100 shown in FIG. 1A. The spectroscopy system 200 includes
a first reservoir 202 that holds one or more emitters (not shown),
such as organic molecules or quantum dots, in solution 11. This
solution 11 flows along an input channel 204 across a photonic
crystal 100 whose energy band structure exhibits a Fano resonance
as described above. A coherent light source 120 illuminates the
particles in the solution 11 and the photonic crystal 100 with an
excitation beam 12 that is resonant with both the photonic
crystal's Fano resonance and the particles. The particles respond
to this excitation by emitting an enhanced emission 13 towards a
detector 130, which generates an electrical signal, such as a
photocurrent, proportional to the intensity of the detected
emission 13. The solution 11 flows off the photonic crystal 100
into an optional second reservoir 208 via an output channel
206.
If desired, the photonic crystal 100, coherent source 120, and 130
may be mounted on translation or rotation stages so that they can
be moved or rotated relative to each other, e.g., to optimize
enhancement of the particles' emission. In addition, the coherent
source 120 may be a tunable source, such as an external-cavity
diode laser or tunable fiber laser, whose output 12 can be tuned on
or off resonance. This tuning can be accomplished with by feeding
back the detector's output to an appropriate control circuit, such
as a proportional-integral-derivative controller or an optimization
circuit.
Consider, for instance, Raman spectroscopy of an organic molecule
using the system 200 shown in FIG. 2A. In Raman spectroscopy,
exciting the molecule at a first frequency causes the molecule to
emit scattered photons--the Raman signal--at a second frequency.
Picking the incident angle to be on-resonance with the pump
wavelength maximizes the excitation enhancement. And picking the
second frequency to match the frequency corresponding to the
photonic crystal's angle of maximum extraction enhancement
maximizes the extraction enhancement. These frequencies, along with
the angle of maximum extraction enhancement, can be determined by:
(1) identifying the angle of maximum extraction enhancement; (2)
determine the frequency corresponding to the angle of maximum
extraction enhancement (e.g., as in the inset of FIG. 8); (3)
selecting the Raman frequency to be on-resonance (i.e.,
substantially equal to the frequency corresponding to the angle of
maximum extraction enhancement); (4) selecting the pump frequency
corresponding to the Raman frequency; and (5) selecting the pump
beam's incidence angle such that the pump frequency is
substantially equal to the frequency of the photonic crystal's Fano
resonance. Illuminating the molecule (and the photonic crystal) at
the resulting incidence angle and pump frequency leads to resonant
pumping and maximum enhancement. Changing the pump beam's incident
angle, frequency, or both away from the values determined according
to this process tunes the pump off-resonance.
Solar Concentrators Based on Photonic Crystal
FIG. 2B illustrates a solar concentrator 250 that includes a
photonic crystal 100 whose energy band structure includes a Fano
resonance at or near the .GAMMA. point. Sunlight 18 illuminates a
frequency converter 252 on top of the photonic crystal 100 over a
range of angles. The frequency converter 252, which may include a
phosphorescent, fluorescent, or nonlinear optical material, up- or
down-converts the incident sunlight to a different frequency. For
instance, the frequency converter 252 may comprise phosphorescent
material that emits radiation over a relatively narrow range of
wavelengths (or even a single wavelength) when illuminated with
sunlight 18 or light from a broadband source.
The frequency-converted sunlight 19 emitted by the frequency
converter 252 propagates into the photonic crystal, which transmits
it to a solar cell 256, such as photovoltaic cell. Optional
reflectors 254 on the photonic crystal's edges couple stray photons
back into the photonic crystal 100. More precisely, the frequency
converter's emission couples to the photonic crystal's resonance(s)
and radiates according to the spectral density of states as
explained above. Because the photonic crystal's spectral density of
states is modified with respect to that of free space, the photonic
crystal 100 transmits the frequency-converted sunlight 19
preferentially in one direction as shown in FIG. 2B. The solar cell
256 is arranged to capture substantially all of this
frequency-converted sunlight 19. As a result, the solar cell 256
operates with relatively high efficiency.
Methods of Fabricating Highly Uniform Photonic Crystals with Fano
Resonances
High quality-factor resonances in a photonic nano-structure depends
on consideration of the structure's bulk material properties and
its sub-wavelength geometry. Material absorption sets the upper
bound of the attainable quality factor, while the structure
geometry can be optimized to minimize Rayleigh scattering due to
surface roughness. One favorable candidate for achieving high
quality factor resonances in the visible portion of the
electromagnetic spectrum is a slab of Si.sub.3N.sub.4 deposited on
top of microns-thick oxide layer of a silicon wafer. With
refractive index of 2.02, Si.sub.3N.sub.4 provides sufficient index
contrast with the SiO.sub.2 below and air or fluid on top. Other
suitable materials include, but are not limited to SiN, TiO.sub.2,
GaAs, and AlGaAs.
The uniformity of the photonic crystal's periodic pattern of holes
also affects the quality factor of the photonic crystal's
resonance(s). In general, the holes' diameters, spacings, and
depths should match as closely as possible over as large an area as
possible for maximum quality factors (and maximum enhancement). In
practice, however, it can be difficult to etch uniform holes over a
large area.
FIG. 3 depicts a system 300 for interference lithography, which is
one technique for fabricating a photonic crystal with a uniform
array of holes extending over a large area of a dielectric
substrate 308. A coherent light source, such as a HeCd laser 302,
emits a beam of radiation. A spatial filter 304 removes undesired
spatial frequency components from this beam so as to create a beam
with a Gaussian profile (or any other suitable profile). This
Gaussian beam illuminates at least a portion of the substrate 308,
which is mounted on a rotation stage 310. The Gaussian beam also
illuminates a mirror 306, which is mounted on the rotation stage
310 at an angle with respect to the substrate 308. Light reflects
off the mirror 306 towards the substrate's surface, where it
interferes with the direct illumination to create a fringe
pattern.
FIGS. 4A-4D illustrate a process for fabricating a photonic crystal
using the interference lithography system 300 shown in FIG. 3. As
shown in FIG. 4A, the interference lithography system 300 projects
a fringe pattern 410 onto the surface of the substrate 308, which
includes a negative photoresist, 408 (e.g., PS-4), an intermediate
layer 406, and an anti-reflection coating 404 (e.g., XHRiC-16), all
layered on a semiconductor slab 302 (e.g., Si.sub.3N.sub.4).
Processing the photoresist 408 with developer, as shown in FIG. 4B,
imparts a pattern corresponding to the fringe pattern 410 on the
photoresist 408. This pattern exposes portions of the underlying
layers, which are etched using reactive ion etching 412, as shown
in FIG. 4C, to create a periodic array of holes in the
semiconductor slab 402. Removing the photoresist 408, intermediate
layer 406, and the anti-reflection coating 404 from the patterned
semiconductor slab 402 yields a completed photonic crystal, as
shown in FIG. 4D.
FIG. 4E shows (a) top-view, (b) tilt-view, and (c) side-view
scanning electron microscope images of a photonic crystal
fabricated according to the process illustrated in FIGS. 4A-4D. The
photonic crystal includes a 250 nm thick slab of SbN.sub.4 with
periodic cylindrical holes on top of 6 .mu.m thick SiO.sub.2 layer.
The holes are spaced at an average period of 320 nm, with an
average hole diameter of 160 nm and an average hole depth of 55 nm.
Uniform periodic patterns were obtained on samples as large as 3
cm.sup.2. These large areas of uniform (defect-free) periodicity
support higher quality factors, which in turn lead to greater
enhancement.
Simulated and Experimental Characterization of an Exemplary
Photonic Crystal
FIGS. 5(a)-5(f) show simulations and experimental measurements of
the performance of the photonic crystal pictured in FIG. 4. FIGS.
5(a) and 5(d) show measured spectral reflectivities for orthogonal
pump polarizations. These reflectivities were obtained by
illuminating the photonic crystal slab with light from a
supercontinuum laser source at small incident angles as measured
from the normal to the photonic crystal plane towards the x-axis.
The reflection spectra in FIGS. 5(a) and 5(d) reveal that the
photonic crystal has eight energy bands.
The measured spectral reflectivities of FIGS. 5(a) and 5(d) show
that the incident beam may excite any one of eight different modes
of the photonic crystal depending on its polarization. (FIGS. 5(b)
and 5(e) each show respective slices of these reflectivity spectra
at an angle of 1.8.degree..) This polarization-dependent excitation
can be understood from symmetry considerations: exciting the
photonic crystal slab with a source of one type of symmetry results
in coupling to the modes of the same type of symmetry only. Moving
away from .GAMMA. to X causes the symmetry group to change from
C.sub.4v to C.sub.1h, reducing the number of irreducible
representations from 5 to 2. Mirror reflection operation around the
x-axis leaves the modes of one irreducible representation
unchanged, while the modes of the other irreducible representation
are altered by a factor of -1.
FIGS. 5(c) and 5(f) show the band diagram, at the .GAMMA. point, of
the photonic crystal's eight lowest energy modes calculated by FDTD
simulations. The four lower frequencies bands are TE-like (numbered
1-4) and the four higher frequencies are TM-like (numbered 5-8).
The E.sub.z component of all eight modes are calculated at the
center of the Si.sub.3N.sub.4 layer at k=[0.01, 0](2 pi/a). Except
for TE-like mode number 2 in FIG. 5(a), the calculated resonant
wavelengths are shifted by no more than .+-.0.5% from the measured
spectra, which is well within the uncertainty of the measured
periodicity or the value of the refractive index. The symmetry of
each mode can be determined by examining the mode profile of its
E.sub.z component. Modes 1, 2, 4, and 6 are altered by a factor -1
under mirror reflection operation around the x-axis and hence
excited by E.sub.y polarized source, while modes 3, 5, 7, and 8 are
unchanged under the same operation and hence excited by E.sub.x
polarized source.
FIG. 6 depicts the calculated Q.sup.total.sub.rad of the eight
bands shown in FIG. 5. It shows that while the doubly-degenerate
(at .GAMMA.) bands 3, 4, 6, and 7 have finite Q.sup.total.sub.rad
at k.apprxeq.0, the singly-degenerate (at .GAMMA.) bands 1, 2, 5,
and 8 have Q.sup.total.sub.rad that go to infinity when approaching
k=0. This can be qualitatively understood from symmetry arguments.
As mentioned earlier, a mode at the .GAMMA. point belongs to one of
five irreducible representations of the C.sub.4v point group. One
of the irreducible representations is doubly degenerate and has the
same symmetry as free space modes, while the rest are singly
degenerate and are decoupled from free-space modes. As a result,
Q.sup.total.sub.rad of these four singly-degenerate modes at the
.GAMMA. point should be infinite despite lying within the light
cone, while the doubly-degenerate modes have finite
Q.sup.total.sub.rad. Moving away from .GAMMA. to X, the point group
becomes C.sub.1h and the doubly-degenerate modes split into two.
The two irreducible representations of the C.sub.1h point group
share symmetry with the free-space modes and therefore
Q.sup.total.sub.rad becomes finite for all resonances, as is
evident from FIG. 6.
A semi-analytical temporal coupled-mode theory model that accounts
for the presence of guided leaky resonances in the Si.sub.3N.sub.4
layer provides a deeper insight into the physics of the measured
resonances. This model can excited with an incident source
propagating from the top and impinging onto the Si.sub.3N.sub.4
layer resonant cavity makes it possible. Applying a first-order
perturbation to Maxwell's equations and energy conservation
considerations, and neglecting second-order effects, yields the
following expression for the photonic crystal's reflectivity:
.tau..tau..gamma..tau..times..tau..gamma..times..function..omega..omega..-
gamma..gamma..tau. ##EQU00022##
.tau..sub.d and .tau..sub.d are the complex reflection and
transmission coefficients of the sample without the square lattice
of cylindrical air holes. .gamma..sub.tol and .gamma..sub.SiO2 are
the coupling strengths of the resonant mode to the top environment
and the SiO.sub.2 layer respectively, and can be related to the
quality factors by
.gamma..sub.SiO2.sup.2=.omega..sub.0/Q.sup.SiO2.sub.rad and
.gamma..sub.tol.sup.2=.omega..sub.0/Q.sup.tol.sub.rad. Equation (5)
shows that there are two temporal pathways: tau.sub.d, which
represents the direct transmission and reflection processes of the
uniform stack, and t.sub.d, which represents the guided resonances
excited within the Si.sub.3N.sub.4 layers whose energy leaks into
the far field. The superposition of the two physical processes
contributes to the typical narrow Fano line shapes superimposed on
a Fabry-Perot-like background as observed in the reflectivity
spectra of FIGS. 5(b) and 5(e).
FIG. 7 is a plot of the total quality factor versus angle obtained
by fitting Eq. (5) to the measured spectra, then using the result
to find the total quality factor,
Q.sub.total(1/Q.sup.total.sub.rad+1/Q.sup.total.sub.loss).sup.-1.
The inset at right in FIG. 7 shows an example of a fitted Fano
resonance curve for the data measured at 0.8.degree. of band 5. The
insets at left show the reflectivity spectra of leaky mode 5 at
angles of 0.1.degree., 0.4.degree., and 0.8.degree.. Note the
distinct higher quality factors of the singly-degenerate modes
close to zero angle (zero wave vector).
FIG. 7 reveals a distinction between the singly degenerate modes
(modes 1, 2, 5, and 8) and the doubly degenerate modes (modes 3, 4,
6 and 7) at small angles. While the measured value of Q.sub.total
increases when approaching k=0 for modes 1, 5, and 8, the doubly
degenerate modes have decreasing or fixed values. Although
Q.sub.total as high as 10.sup.4 are observed, the calculated
Q.sup.total.sub.rad (FIG. 6) of the singly-degenerate modes are
much greater at small angles, suggesting that close to k=0 the
resonant energy decay is dominated by absorption and incoherent
scattering from fabrication imperfections
(Q.sup.total.about.Q.sup.total.sub.loss.about.10.sup.4), both of
which could be significantly reduced by improving the fabrication
process.
On the other hand, the four low-Q bands (modes 3, 4, 6, and 7) in
FIG. 7 have Q.sup.total values that are comparable to the
calculated Q.sup.total.sub.rad and smaller than
Q.sup.total.sub.loss. Indeed, FDTD calculations of the resonant
mode show that the energy confinement is approximately unchanged
within the plotted range of angles, suggesting that
Q.sup.scat.sub.loss is relatively constant over the range of angles
considered here. Apart from limiting the values of Q.sup.total and
hence the linewidth of the resonant lineshapes, the presence of
relatively large scattering loss and absorption compared to
far-field radiation near normal incidence leads to reduced resonant
amplitudes. Conversely, the decrease of Q.sup.total.sub.rad away
from the normal provides a better match between Q.sup.scat.sub.rad
and Q.sup.total.sub.rad, which leads to an increase in the
features' height. This is consistent with Eq. (5), and also
explains why band 2 appears only weakly in the measurement results
shown in FIG. 5(a). Unlike other high Q.sup.total.sub.rad modes
whose values decrease rapidly away from the .GAMMA. point, the
Q.sup.total.sub.rad of the missing TE-like band 2 remains high
(FIG. 6) for most angles, resulting in small reflectivity
amplitudes which are harder to detect.
Experimental realization of this mode offers several possible
advantages: (1) the strongly enhanced field close to the photonic
crystal surface and the simple access to it provides a new platform
for the study of light and matter interaction; (2) it offers an
easy-to-fabricate structure that supports delocalized modes with
ultrahigh quality factors; (3) it allows a simple coupling of
external radiation to strongly confined modes; and (4) despite the
macroscopically large area resonator, only a few high-Q modes are
supported within a fairly broad frequency range. The realization of
this novel resonance could enable the enhancement and the
demonstration of new physical phenomena in biological sensing,
laser physics, energy conversion, nonlinear optics, and optical
filters.
Experimental Demonstration of Enhanced Fluorescence Emission and
Lasing
FIGS. 8-10 show theoretical predictions and experimental
measurements of the performance of an organic molecule laser. This
laser includes an organic molecule (e.g., a Rhodamine 6G molecule)
on the surface of a photonic crystal with a Fano resonance at or
near the .GAMMA. point (e.g., the photonic crystal 100 shown in
FIG. 1). A pump source--here, another laser--illuminates the
organic molecule in a first direction, causing the organic molecule
to fluoresce preferentially in a second direction as explained
above. A spectrometer with a resolution of about 0.03 nm (e.g., an
Ocean Optics HR4000) collects the fluorescence spectrum. Changing
the spectrometer's position with respect to the organic molecule
(e.g., using an XYZ translation stage) makes it possible to detect
fluorescence emitted into different angles along the .GAMMA.-X and
.GAMMA.-M directions.
FIG. 8 is a plot of the emission spectra from a blank photonic
crystal slab, Rhodamine 6G molecules in solution on the photonic
crystal slab illuminated by a non-resonant pump beam, and the same
Rhodamine 6G solution illuminated by a resonant pump beam. The
blank slab's emission spectrum is fairly flat and low. The
non-resonantly pumped Rhodamine 6G molecule's emission spectrum has
small peaks at about 577 nm and about 580 nm. And the resonantly
pumped Rhodamine 6G molecule's emission spectrum has large peaks at
about 574 nm, 577 nm, and 580 nm. In the on-resonance case, the
excitation field within d.sup.pump.sub.eff from the surface is
strongly enhanced compared to the off-resonance case, while the
remainder of the bulk layer exhibits no enhancement. In fact, FIG.
8 shows that resonant pumping enhances the Rhodamine 6G molecule's
emission spectrum by a factor of at least about 80, which agrees
well with the predicted enhancement.
The inset of FIG. 8 is a plot of the photonic crystal slab's band
structure obtained from an FDTD. It shows that the incident angle
for on-resonance coupling is about 10.0.degree. for an excitation
wavelength of 532 nm. Experiments show that the incident angle for
on-resonance coupling is about 10.02.degree. for this excitation
wavelength, showing good agreement with the calculated results.
As explained above, the pump can be tuned on- and off-resonance by
(1) changing the pump beam's incident angle, (2) changing the pump
beam's wavelength, and (3) by changing both the incident angle and
the wavelength. In these experiments, off-resonance pumping was
achieved by changing the pump beam's incident angle. As a result,
the difference in the enhancement for on- and off-resonance pumping
is due to excitation enhancement since the extraction enhancement
for the same wavelength at the same emission angle remains the
same.
FIG. 9 includes plots of predicted and calculated enhancement data.
FIG. 9(a) is a close-up of the plot of the photonic crystal's
energy band structure along the .GAMMA. to M and .GAMMA. to X
directions shown in the inset of FIG. 8. It shows the five energy
bands in the wavelength/angular region of interest. FIG. 9(b) is a
plot of angle-resolved fluorescence measurements of the Rhodamine
6G solution suspended on top of the PhC. As expected, it mimics the
photonic crystal's energy band structure. FIG. 9(b) shows that the
fluorescence is maximized around the photonic crystal's resonances.
This can be understood by considering the decay rate into
frequency, which is maximized at .omega.=.omega..sub.k. FIG. 9(b)
also shows the emission's strong angular dependence.
FIGS. 9(c), 9(d), and 9(e) are plots of the total enhancement
versus angle for energy bands 1 and 4 in FIGS. 9(a) and 9(b). The
data plotted in FIG. 9(c) is the product of excitation enhancement
and extraction enhancement using the theoretical model discussed
above. These data show that the total enhancement goes to zero for
both bands at .GAMMA. since the bands' radiative quality factors
are infinite at .GAMMA.. Away from .GAMMA., the bands' radiative
quality factors drop exponentially, leading to maximum total
enhancement upon satisfaction of the Q-matching condition between
the radiative and non-radiative quality factors. The radiative
quality factor of mode 1 drops much more slowly than that of mode
4, which may explain why the total enhancement of mode 1 increases
much more slowly than that of mode 4 near F.
FIG. 9(d) shows the predicted averaged total enhancement factor
between 0.degree. and 1.5.degree.. This plot represents the
predicted total enhancement is averaged over the range of wave
vectors corresponding to the spectrometer aperture's acceptance
angle. This angle is narrow in the x direction (the difference
between resonances of allowed wave vectors within corresponding
acceptance angle is small compared to resonance width), but wide
(e.g., about 1.degree.) in the y direction.
Limitations on quality factor measurements restrict the calculation
range for the averaged total enhancement to about 0-1.5.degree.
along the .GAMMA.-X direction. FIG. 9(d) shows that bands 1 and 4
behave differently close to the .GAMMA. point: for mode 4, the
Q-matching condition is satisfied close to .GAMMA. (around
0.5.degree.) and the averaged total enhancement is almost constant
near .GAMMA.; however, the Q-matching for mode 1 occurs far from
.GAMMA., resulting in a linear increase in the averaged total
enhancement at small angles.
FIG. 9(e) shows the total enhancement factor extracted from
experimental results in FIG. 9(b). While the experimentally
determined total enhancement of band 4 remains almost constant near
.GAMMA. and drops to 0 when the resonance falls out of the range of
spectrometer, the experimentally determined total enhancement of
band 1 increases with angle before reaching its maximum of
6.3.times.10.sup.3 at 3.degree.. These experimental observations
agree fairly well with the theoretical predictions in FIG. 9(d),
particularly if comparing the theoretical prediction of the maximum
enhancement for band 1 to experiments. This agreement holds for
both trends and values: e.g., for band 4 at .GAMMA., the averaged
total enhancement is about 3.times.10.sup.3 compared to an
experimentally determined value of 2.8.times.10.sup.3.
FIG. 10 shows predictions and results of a lasing experiment
carried out using the same 100 nm layer of Rhodamine 6G solution,
photonic crystal, and spectrometer as in the fluorescence
measurements. In the lasing experiment, however, the Rhodamine 6G
was pumped with a pulsed second-harmonic beam from a Nd:YAG laser.
The pump beam had a wavelength of 532 nm, a pulse duration of 5 ns,
a pulse repetition rate of 10 Hz, and a beam diameter of 1 mm.
Pumping the Rhodamine 6G solution lased in two modes: a first mode
(mode 4 in FIG. 9) at about 580 nm lased first, followed by a
second mode (mode 1 in FIG. 9) at about 575 nm, both well within
the Rhodamine 6G's emission spectrum. The total quality factor of
mode 4 was about 8.3.times.10.sup.3.
The solid lines in the main plot and lower inset of FIG. 10 are
analytic predictions of the laser's output energy versus the pump
energy on logarithmic and linear scales, respectively. Red circles
are energies measured using the spectrometer. Green circles are
data measured with a power meter. The jump in output power
indicates the onset of lasing. The lower inset shows that the
output power grows linearly with the pump energy beyond threshold.
The upper inset is the measured power spectrum of emission from the
photonic crystal slab at normal incidence below and above the
lasing threshold.
FIG. 10 shows that the theoretical predictions of both threshold
and slope efficiency match reasonably well with the experimental
results within experimental errors. In particular, the measured
threshold energy is 9 .mu.J/cm.sup.3 corresponding to the intensity
of 1.8 kW/cm.sup.2, which is very low. Without being bound by any
particular theory, it appears that the threshold is reduced for at
least three reasons (1) the high-Q factors of the special Fano
resonances; (2) the substantially enhanced absorption of the pump
within 100 nm thin layer enabled by the excitation enhancement; and
(3) the enhancement of the spontaneous emission factor for the
lasing mode due to the enhanced spectral density of states. In
particular, the excitation enhancement of about 60 enables 12.7%
absorption of the pump energy within only 100 nm thin layer of the
dye solution. And the rate of spontaneous emission into the
structure's lasing mode is higher than in free space, yielding a
higher spontaneous emission factor.
CONCLUSION
While various inventive embodiments have been described and
illustrated herein, those of ordinary skill in the art will readily
envision a variety of other means and/or structures for performing
the function and/or obtaining the results and/or one or more of the
advantages described herein, and each of such variations and/or
modifications is deemed to be within the scope of the inventive
embodiments described herein. More generally, those skilled in the
art will readily appreciate that all parameters, dimensions,
materials, and configurations described herein are meant to be
exemplary and that the actual parameters, dimensions, materials,
and/or configurations will depend upon the specific application or
applications for which the inventive teachings is/are used. Those
skilled in the art will recognize, or be able to ascertain using no
more than routine experimentation, many equivalents to the specific
inventive embodiments described herein. It is, therefore, to be
understood that the foregoing embodiments are presented by way of
example only and that, within the scope of the appended claims and
equivalents thereto, inventive embodiments may be practiced
otherwise than as specifically described and claimed. Inventive
embodiments of the present disclosure are directed to each
individual feature, system, article, material, kit, and/or method
described herein. In addition, any combination of two or more such
features, systems, articles, materials, kits, and/or methods, if
such features, systems, articles, materials, kits, and/or methods
are not mutually inconsistent, is included within the inventive
scope of the present disclosure.
The above-described embodiments can be implemented in any of
numerous ways. For example, the embodiments may be implemented
using hardware, software or a combination thereof. When implemented
in software, the software code can be executed on any suitable
processor or collection of processors, whether provided in a single
computer or distributed among multiple computers.
Further, it should be appreciated that a computer may be embodied
in any of a number of forms, such as a rack-mounted computer, a
desktop computer, a laptop computer, or a tablet computer.
Additionally, a computer may be embedded in a device not generally
regarded as a computer but with suitable processing capabilities,
including a Personal Digital Assistant (PDA), a smart phone or any
other suitable portable or fixed electronic device.
Also, a computer may have one or more input and output devices.
These devices can be used, among other things, to present a user
interface. Examples of output devices that can be used to provide a
user interface include printers or display screens for visual
presentation of output and speakers or other sound generating
devices for audible presentation of output. Examples of input
devices that can be used for a user interface include keyboards,
and pointing devices, such as mice, touch pads, and digitizing
tablets. As another example, a computer may receive input
information through speech recognition or in other audible
format.
Such computers may be interconnected by one or more networks in any
suitable form, including a local area network or a wide area
network, such as an enterprise network, and intelligent network
(IN) or the Internet. Such networks may be based on any suitable
technology and may operate according to any suitable protocol and
may include wireless networks, wired networks or fiber optic
networks.
The various methods or processes outlined herein may be coded as
software that is executable on one or more processors that employ
any one of a variety of operating systems or platforms.
Additionally, such software may be written using any of a number of
suitable programming languages and/or programming or scripting
tools, and also may be compiled as executable machine language code
or intermediate code that is executed on a framework or virtual
machine.
In this respect, various inventive concepts may be embodied as a
computer readable storage medium (or multiple computer readable
storage media) (e.g., a computer memory, one or more floppy discs,
compact discs, optical discs, magnetic tapes, flash memories,
circuit configurations in Field Programmable Gate Arrays or other
semiconductor devices, or other non-transitory medium or tangible
computer storage medium) encoded with one or more programs that,
when executed on one or more computers or other processors, perform
methods that implement the various embodiments of the invention
discussed above. The computer readable medium or media can be
transportable, such that the program or programs stored thereon can
be loaded onto one or more different computers or other processors
to implement various aspects of the present invention as discussed
above.
The terms "program" or "software" are used herein in a generic
sense to refer to any type of computer code or set of
computer-executable instructions that can be employed to program a
computer or other processor to implement various aspects of
embodiments as discussed above. Additionally, it should be
appreciated that according to one aspect, one or more computer
programs that when executed perform methods of the present
invention need not reside on a single computer or processor, but
may be distributed in a modular fashion amongst a number of
different computers or processors to implement various aspects of
the present invention.
Computer-executable instructions may be in many forms, such as
program modules, executed by one or more computers or other
devices. Generally, program modules include routines, programs,
objects, components, data structures, etc. that perform particular
tasks or implement particular abstract data types. Typically the
functionality of the program modules may be combined or distributed
as desired in various embodiments.
Also, data structures may be stored in computer-readable media in
any suitable form. For simplicity of illustration, data structures
may be shown to have fields that are related through location in
the data structure. Such relationships may likewise be achieved by
assigning storage for the fields with locations in a
computer-readable medium that convey relationship between the
fields. However, any suitable mechanism may be used to establish a
relationship between information in fields of a data structure,
including through the use of pointers, tags or other mechanisms
that establish relationship between data elements.
Also, various inventive concepts may be embodied as one or more
methods, of which an example has been provided. The acts performed
as part of the method may be ordered in any suitable way.
Accordingly, embodiments may be constructed in which acts are
performed in an order different than illustrated, which may include
performing some acts simultaneously, even though shown as
sequential acts in illustrative embodiments.
All definitions, as defined and used herein, should be understood
to control over dictionary definitions, definitions in documents
incorporated by reference, and/or ordinary meanings of the defined
terms.
The indefinite articles "a" and "an," as used herein in the
specification and in the claims, unless clearly indicated to the
contrary, should be understood to mean "at least one."
The phrase "and/or," as used herein in the specification and in the
claims, should be understood to mean "either or both" of the
elements so conjoined, i.e., elements that are conjunctively
present in some cases and disjunctively present in other cases.
Multiple elements listed with "and/or" should be construed in the
same fashion, i.e., "one or more" of the elements so conjoined.
Other elements may optionally be present other than the elements
specifically identified by the "and/or" clause, whether related or
unrelated to those elements specifically identified. Thus, as a
non-limiting example, a reference to "A and/or B", when used in
conjunction with open-ended language such as "comprising" can
refer, in one embodiment, to A only (optionally including elements
other than B); in another embodiment, to B only (optionally
including elements other than A); in yet another embodiment, to
both A and B (optionally including other elements); etc.
As used herein in the specification and in the claims, "or" should
be understood to have the same meaning as "and/or" as defined
above. For example, when separating items in a list, "or" or
"and/or" shall be interpreted as being inclusive, i.e., the
inclusion of at least one, but also including more than one, of a
number or list of elements, and, optionally, additional unlisted
items. Only terms clearly indicated to the contrary, such as "only
one of" or "exactly one of," or, when used in the claims,
"consisting of," will refer to the inclusion of exactly one element
of a number or list of elements. In general, the term "or" as used
herein shall only be interpreted as indicating exclusive
alternatives (i.e. "one or the other but not both") when preceded
by terms of exclusivity, such as "either," "one of," "only one of,"
or "exactly one of" "Consisting essentially of," when used in the
claims, shall have its ordinary meaning as used in the field of
patent law.
As used herein in the specification and in the claims, the phrase
"at least one," in reference to a list of one or more elements,
should be understood to mean at least one element selected from any
one or more of the elements in the list of elements, but not
necessarily including at least one of each and every element
specifically listed within the list of elements and not excluding
any combinations of elements in the list of elements. This
definition also allows that elements may optionally be present
other than the elements specifically identified within the list of
elements to which the phrase "at least one" refers, whether related
or unrelated to those elements specifically identified. Thus, as a
non-limiting example, "at least one of A and B" (or, equivalently,
"at least one of A or B," or, equivalently "at least one of A
and/or B") can refer, in one embodiment, to at least one,
optionally including more than one, A, with no B present (and
optionally including elements other than B); in another embodiment,
to at least one, optionally including more than one, B, with no A
present (and optionally including elements other than A); in yet
another embodiment, to at least one, optionally including more than
one, A, and at least one, optionally including more than one, B
(and optionally including other elements); etc.
In the claims, as well as in the specification above, all
transitional phrases such as "comprising," "including," "carrying,"
"having," "containing," "involving," "holding," "composed of," and
the like are to be understood to be open-ended, i.e., to mean
including but not limited to. Only the transitional phrases
"consisting of" and "consisting essentially of" shall be closed or
semi-closed transitional phrases, respectively, as set forth in the
United States Patent Office Manual of Patent Examining Procedures,
Section 2111.03.
* * * * *